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%Notations Adrien

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\begin{document}

27th IFIP TC7 Conference 2015  on System Modelling and Optimization\\

Modelling and Control in Contact Mechanics

\begin{center}



\vspace{.4in}

{\bf Existence and Uniqueness Results for Elasto--plastic Contact Problems \\
with Heat Transfer}

\vspace{.3in}

{\bf  Adrien Petrov}


\vspace{.3in}

INSA--Lyon, Institut Camille Jordan CNRS UMR 5208, France


\vspace{.2in}

apetrov@math.univ-lyon1.fr\\


\vspace{.3in}


\end{center}
{\bf Abstract:}
We consider an elasto--plastic bar of length \(L\) vibrating longitudinally. This
bar is free to move on the one end as long as it does not hit an elasto--plastic obstacle
while on the other end a force is applied.
The problem for the unknown displacement and temperature is rewritten in accordance
with the formalism of hysteresis operators.
Let \(u(x, t)\) be the displacement at time \(t\) of the material
point of spatial coordinate \(x\in(0,L)\).
The motion is governed by the equation
\begin{equation}
\label{eq:1}
\rho u_{tt} - \sigma_x=0,
\end{equation}
where \((\cdot)_t\eqldef\frac{\partial(\cdot)}{\partial t}\)
and
\((\cdot)_x\eqldef\frac{\partial(\cdot)}{\partial x}\) and \(\rho>0\)
denotes the mass density.
The stress tensor $\sigma$ is assumed to satisfy the constitutive equation
\(\sigma \eqldef \calp[\ve] + \nu \ve_t - \beta(\theta - \thre)\)
where $\ve\eqldef  u_x$,
$\theta(x,t) > 0$ is the absolute temperature which is one
of the unknowns of the problem, $\nu > 0$ is the viscosity modulus,
$\beta \in \real$ is the thermal expansion coefficient, and $\thre > 0$
is a given referential temperature.
The symbol \(\calp[\ve] = \lambda \ve + \calp_0[\ve]\) denotes a constitutive
operator of elasto--plasticity where $\calp_0[\ve]$ corresponds to the plastic stress
component with yield point $r>0$
and elasticity domain $K\eqldef[-r,r]$ and satisfies
\be{varin3sc}
\begin{cases}
\calp_0[\ve](t) \in K & \text{for all }t \in [0,T],\\
\calp_0[\ve](0) = \mathrm{Proj}_{K}(\mathbb{E}\ve(0)),\\
(\mathbb{E}\ve_t(t) {-} \calp_{0,t}[\ve](t))(
\calp_0[\ve](t){-} y) \ge 0 & \text{a.e. for all } y \in K,
\end{cases}
\ee
where \(\mathrm{Proj}_{K}\) is the projection onto \(K\),
$\mathbb{E}>0$ is the elasticity modulus and
$\lambda > 0$ is the kinematic hardening modulus.

\vspace{0.5em}
A space discretization is introduced and some a priori estimates are
obtained by using both the classical energy estimate and more specific techniques like
the Dafermos estimate and some Sobolev interpolation inequalities.
Therefore the existence result is derived. The uniqueness result
follows from the Lipschitz continuity of the nonlinearities.

\vspace{0.5em}
This is a joint work with Pavel Krej\v{c}\'{i}.

\end{document}
